Relationship between hyperbolic cosine and cosine

I am considering the hyperbola [itex]x^2-y^2=1[/itex] and its intersection with the line y=mx. The positive x-coordinate of the intersection is given by: [tex]x=\sqrt{\frac{1}{1-\tan^2\alpha}}=\sqrt{\frac{\cos^2 \alpha}{\cos(2\alpha)}}=\cos\alpha \sqrt{\sec(2\alpha)}[/tex] where we used the identity [itex]m=\tan\alpha[/itex].

However, using Euler formulas for cosines does not seem to give the relationship: [itex]\cosh(\alpha)=\cos(i\alpha)[/itex].
Am I using a wrong geometrical definition of hyperbolic cosine? I mean, perhaps the hyperbolic cosine is not simply the x-coordinate of the intersection of a ray with the hyperbola?

I am considering the hyperbola [itex]x^2-y^2=1[/itex] and its intersection with the line y=mx. The positive x-coordinate of the intersection is given by: [tex]x=\sqrt{\frac{1}{1-\tan^2\alpha}}=\sqrt{\frac{\cos^2 \alpha}{\cos(2\alpha)}}=\cos\alpha \sqrt{\sec(2\alpha)}[/tex] where we used the identity [itex]m=\tan\alpha[/itex].

However, using Euler formulas for cosines does not seem to give the relationship: [itex]\cosh(\alpha)=\cos(i\alpha)[/itex].
Am I using a wrong geometrical definition of hyperbolic cosine? I mean, perhaps the hyperbolic cosine is not simply the x-coordinate of the intersection of a ray with the hyperbola?

Yes, you are. The line y= mx has nothing to do with it. 'cos(t)' is defined as the x-coordinate of the point (x,y) at distance t around the circumference of the circle, [itex]x^2+ y^2= 1[/itex] from (1, 0).

So 'cosh(t)' is the x-coordinate of (x, y) at distance t around the curve [itex]x^2- y^2= 1[/itex] from (1, 0).